Part 1: Algebraic Structures with One Binary Operation

1. Semi Groups, Monoids, and Groups

1.1 Semi Groups

1. Define a semi group.
2. Provide an example of a semi group that is not a monoid.

1.2 Monoids 3. Define a monoid. 4. Give an example of a monoid and explain why it satisfies the monoid properties.

1.3 Groups 5. Define a group. 6. Prove that the set of integers under addition forms a group.

2. Congruence Relation and Quotient Structures

2.1 Congruence Relation 7. Define a congruence relation on a group. 8. Give an example of a congruence relation on the group of integers under addition.

2.2 Quotient Structures 9. Define the quotient group. 10. For the group of integers under addition, consider the congruence relation modulo 4. List the elements of the quotient group.

Part 2: Free and Cyclic Monoids and Groups

3. Free and Cyclic Monoids

3.1 Free Monoids 11. Define a free monoid. 12. Give an example of a free monoid generated by a set with two elements.

3.2 Cyclic Monoids 13. Define a cyclic monoid. 14. Provide an example of a cyclic monoid and describe its generator.

4. Free and Cyclic Groups

4.1 Free Groups 15. Define a free group. 16. Provide an example of a free group with a single generator.

4.2 Cyclic Groups 17. Define a cyclic group. 18. Show that every subgroup of a cyclic group is cyclic.

Part 3: Permutation Groups, Substructures, and Normal Subgroups

5. Permutation Groups

1. Define a permutation group.